AMC 12 AMC 12 Trigonometry & Complex Numbers 2 — Questions and Answers
Question 1: What is the product (2 + 3i)(2 − 3i)?
- 4
- 13 (Correct answer)
- −5
- 4 + 9i
Correct answer: 13
(2 + 3i)(2 − 3i) = 4 − 9i² = 4 + 9 = 13.
Question 2: In triangle ABC, angle A = 30°, b = 6, c = 4. What is the area of the triangle?
- 3
- 6 (Correct answer)
- 12
- 4√3
Correct answer: 6
Area = (1/2)bc·sin(A) = (1/2)(6)(4)(1/2) = 6.
Question 3: What is tan(π/4)?
- 0
- √3
- 1 (Correct answer)
- 1/2
Correct answer: 1
tan(π/4) = sin(π/4)/cos(π/4) = (√2/2)/(√2/2) = 1.
Question 4: What is the real part of (1 + 2i)/(1 − i)?
- 1/2
- −1/2 (Correct answer)
- 3/2
- 1
Correct answer: −1/2
Multiplying by the conjugate: (1+2i)(1+i)/2 = (−1+3i)/2, giving real part −1/2.
Question 5: Which expression correctly states the sine angle-addition formula?
- sin A·cos B + cos A·sin B (Correct answer)
- sin A·sin B + cos A·cos B
- cos A·cos B − sin A·sin B
- sin A·cos B − cos A·sin B
Correct answer: sin A·cos B + cos A·sin B
The angle-addition identity is sin(A + B) = sin A·cos B + cos A·sin B.
Question 6: What is |e^(iπ) + 1|?
- 0 (Correct answer)
- 1
- 2
- π
Correct answer: 0
By Euler's formula e^(iπ) = −1, so e^(iπ) + 1 = 0, and |0| = 0.
AMC 12 AMC 12 Trigonometry & Complex Numbers 2
What is the product (2 + 3i)(2 − 3i)?